Noncentral F Distribution Functions

Functions to compute the probability density function, cumulative distribution function, and quantile function for the Noncentral F distribution.

The noncentral F distribution is a generalization of the Fisher F Distribution.

Thje PDF is:

$$f(x;\nu_1,\nu_2, \lambda)=\sum \limits_{k=0}^{\infty }{\frac {e^{-\lambda /2}(\lambda /2)^{k}}{B\left({\frac {\nu_{2}}{2}},{\frac {\nu_{1}}{2}}+k\right)k!}}\left({\frac {\nu_{1}}{\nu_{2}}}\right)^{{\frac {\nu_{1}}{2}}+k}\left({\frac {\nu_{2}}{\nu_{2}+\nu_{1}x}}\right)^{{\frac {\nu_{1}+\nu_{2}}{2}}+k}x^{\nu_{1}/2-1+k}$$

The CDF is:

$$F(x; d_{1},d_{2},\lambda )=\sum \limits_{j=0}^{\infty }\left({\frac {\left({\frac {1}{2}}\lambda \right)^{j}}{j!}}e^{-\lambda /2}\right)I\left({\frac {d_{1}x}{d_{2}+d_{1}x}}{\bigg |}{\frac {d_{1}}{2}}+j,{\frac {d_{2}}{2}}\right)$$

Usage

non_central_f_distribution(df1, df2, lambda)

non_central_f_pdf(x, df1, df2, lambda)

non_central_f_lpdf(x, df1, df2, lambda)

non_central_f_cdf(x, df1, df2, lambda)

non_central_f_lcdf(x, df1, df2, lambda)

non_central_f_quantile(p, df1, df2, lambda)

Arguments

df1

degrees of freedom for the numerator (df1 > 0)

df2

degrees of freedom for the denominator (df2 > 0)

lambda

noncentrality parameter (lambda >= 0)

x

quantile

p

probability (0 <= p <= 1)

Value

A single numeric value with the computed probability density, log-probability density, cumulative distribution, log-cumulative distribution, or quantile depending on the function called.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Noncentral F distribution with df1 = 10, df2 = 10 and noncentrality
# parameter 1
dist <- non_central_f_distribution(10, 10, 1)
# Apply generic functions
cdf(dist, 0.5)
#> [1] 0.1142528
logcdf(dist, 0.5)
#> [1] -2.169342
pdf(dist, 0.5)
#> [1] 0.5754471
logpdf(dist, 0.5)
#> [1] -0.552608
hazard(dist, 0.5)
#> [1] 0.6496742
chf(dist, 0.5)
#> [1] 0.1213237
mean(dist)
#> [1] 1.375
median(dist)
#> [1] 1.10075
mode(dist)
#> [1] 0.7349843
range(dist)
#> [1]  0.000000e+00 1.797693e+308
quantile(dist, 0.2)
#> [1] 0.6363179
standard_deviation(dist)
#> [1] 1.063113
support(dist)
#> [1]  0.000000e+00 1.797693e+308
variance(dist)
#> [1] 1.130208
skewness(dist)
#> [1] 3.609738
kurtosis(dist)
#> [1] 51.0825
kurtosis_excess(dist)
#> [1] 48.0825

# Convenience functions
non_central_f_pdf(1, 5, 2, 1)
#> [1] 0.3051418
non_central_f_lpdf(1, 5, 2, 1)
#> [1] -1.186979
non_central_f_cdf(1, 5, 2, 1)
#> [1] 0.3737987
non_central_f_lcdf(1, 5, 2, 1)
#> [1] -0.9840377
non_central_f_quantile(0.5, 5, 2, 1)
#> [1] 1.507635